Question

Dear Experts,

This question correlates to the theory of complex numbers and their usage in algebra. It has come to my attention that I have some difficulty applying general algebra rules to complex numbers. My next exam is focused on Control Theory, where Partial Fractions is used to derive the equation necessary to determine the inverse Laplace of the transfer function. This will then give the equation to analyze the response of the transfer function over time.

I do understand the theory behind the calculations, such as Laplace Transform, Inverse Laplace, Partial Fractions, etc, but when I solve the denominator using the ABC-formula and get a complex number, to perform partial fractions becomes unclear. It is as if I don't know how to further simplify the equation after some point.

I know I can arrange A, B, and C in matrix form to determine their values. But then it becomes difficult to complete the calculations. If someone could provide an explanation, I might be able to understand what the idea behind the calculations is.

Is the approach in the step above correct? And if so, the next step is where it gets complicated to complete the calculations. Especially to determine the value of A and B, which consist of complex numbers. So how do I complete the calculations below?

The results for A, B and C should be:

$\small C = \frac{5}{4}$

The help provided is highly appreciated as it is in preparation for my exams. Thank you in advance for answering my question.

/(s22s+4) (s13)(s +1 - 3) B + (s1 3) (s+1+v3) A(s(s1v3j)) B(s(s+1- V3j)) +C(s2+2s+4) 5 A(s2(s1+ v3i)s)B(s2(s+1- v3j)s) +C(s22s + 4) 5 s2ABC 0 :(1v3i)A (1 V3,)B+2C 0 .: 4C=5 5 5 A = 8 24 5 B = 24

Answer 1

E+25+4) S+3 we Know Thar ati (a-ib S4)*+(B (s+1+8 (41-Bj) (s13S-Rj B A St1- 13j (s+-s(sH+hi-A) A (St1tjs + B (54-s j (4 -)+c(&j254) t S o T Side Gicints Campaing On on A+BtC O 0 F ompetin Gffi aent A bo TE Sid Constant tom an C. ら-4C ArtB ARB-C .fon AB tBj (A -B A -5i |2

AB A B 53i 12 RA A-5 24 -S A+B Fromm 4 34 t. 24